From [1]}, we know that \(\mathcal {S}\)  is attained by the Aubin-Talanti bubbles
\(U_{\varepsilon }(x) :=\frac{2\sqrt{2}\varepsilon }{\varepsilon ^2+|x|^2}, \quad \varepsilon > 0, \quad x\in \mathbb {R}^4.\) 
